Optical sensor having a non-negligible source coherence length

ABSTRACT

An optical sensor, a method of configuring an optical sensor, and a method of using an optical sensor are provided. The optical sensor includes an optical waveguide having a length and a laser source optically coupled to the waveguide. The laser source has a coherence length. Light from the source is transmitted to the waveguide as a first signal propagating along the waveguide in a first direction and a second signal propagating along the waveguide in a second direction opposite to the first direction. The optical paths of the first signal and the second signal are substantially reciprocal with one another and the first signal and the second signal are combined together after propagating through the waveguide to generate a third signal. The coherence length is greater than 1 meter or is in a range between 200 microns and 10 centimeters.

CROSS-REFERENCE TO RELATED APPLICATIONS

The present application is a continuation of U.S. patent applicationSer. No. 12/767,643, filed on Apr. 26, 2010 and incorporated in itsentirety by reference herein, which claims the benefit of priority toU.S. Provisional Patent Application No. 61/173,571, filed Apr. 28, 2009,which is incorporated in its entirety by reference herein. U.S. patentapplication Ser. No. 12/767,643 is also a continuation-in-part of U.S.patent application Ser. No. 12/271,760, filed Nov. 14, 2008, now U.S.Pat. No. 7,911,619 (issued Mar. 22, 2011), and incorporated in itsentirety by reference herein, which claims the benefit of priority toU.S. Provisional Patent Application No. 60/988,404, filed Nov. 15, 2007and incorporated in its entirety by reference herein.

BACKGROUND

1. Field of the Invention

The present application relates generally to optical gyroscopes, andmore specifically, to optical gyroscopes utilizing a laser source.

2. Description of the Related Art

First demonstrated in the 1970s, the fiber-optic gyroscope (FOG) is oneof the oldest and most advanced fiber optic sensors. It has found manycommercial and military applications. Part of this success is rooted inthe Sagnac loop that it uses as the sensing element, a common-pathinterferometer that is inherently reciprocal and thus highly stableagainst most external perturbations. Another component of this successis the use of broadband light, for example from an Er-dopedsuperfluorescent fiber source (SFS), to interrogate the interferometer.Incoherent light was shown very early on to essentially eliminate twodeleterious effects that occur in the fiber loop, namely anon-reciprocal phase drift induced by the nonlinear Kerr effect andnoise and long-term phase drift caused by coherent backscattering. See,e.g., H. C. Lefevre, The Fiber-Optic Gyroscope, Artech House, Boston(1993).

Unfortunately, the adoption of a broadband source introduced twolimitations. First, the noise of a broadband source (excess noise)typically far exceeds shot noise, and it limits the FOG's minimumdetectable rotation rate. Second, the scale factor of a FOG, whichrelates the measured gyroscope signal to the rotation rate, must beextremely stable (˜1 part per million) for aircraft inertial navigationapplications. Consequently, the mean wavelength of the light must have acomparable stability, which is difficult to achieve in practice with abroadband source.

SUMMARY

In certain embodiments, a fiber-optic sensor comprises an optical fibercoil having a length and a laser source optically coupled to the coil.The laser source has a coherence length. Light from the source istransmitted to the coil as a first signal propagating along the coil ina first direction and a second signal propagating along the coil in asecond direction opposite to the first direction. The optical paths ofthe first signal and the second signal are substantially reciprocal withone another and the first signal and the second signal are combinedtogether after propagating through the coil to generate a third signal.The coherence length is greater than 1 meter or is in a range between200 microns and 10 centimeters.

In certain such embodiments, the coherence length is in a range between100 microns and 10 centimeters, in a range between 100 microns and 5centimeters, in a range between 100 microns and 1 centimeter, in a rangebetween 200 microns and 10 centimeters, in a range between 200 micronsand 5 centimeters, in a range between 200 microns and 1 centimeter, in arange between 500 microns and 10 centimeters, in a range between 500microns and 5 centimeters, in a range between 500 microns and 1centimeter, in a range between 1 millimeter and 10 centimeters, in arange between 1 millimeter and 5 centimeters, in a range between 1millimeter and 1 centimeter, in a range between 1 centimeter and 10centimeters, in a range between 1 centimeter and 5 centimeters, greaterthan 1 meter, in a range between 1 meter and 1 kilometer, in a rangebetween 1 meter and 500 meters, in a range between 1 meter and 100meters, in a range between 10 meters and 100 meters, in a range between10 meters and 500 meters, in a range between 10 meters and 1 kilometer,in a range between 100 meters and 500 meters, in a range between 100meters and 1 kilometer, in a range between 100 meters and 10 kilometers,in a range between 1 kilometer and 10 kilometers, in a range between 10kilometers and 100 kilometers, or in a range between 1 kilometer and 100kilometers. In certain such embodiments, the coherence length is lessthan or equal to the length of the coil 20, while in certain otherembodiments, the coherence length is greater than the length of the coil20. The coherence length is selected in certain such embodiments toprovide a noise level below a desired level, and the selection of thecoherence length is based on calculations as described more fullyherein.

In certain embodiments, the coherence length is less than the length. Incertain embodiments, the sensor has a phase noise which varies as afunction of coherence length, the phase noise having a peak value at apredetermined value of coherence length, wherein the phase noise for thecoherence length is at least a factor of two less than the peak value ofthe phase noise. In certain embodiments, the sensor has a phase noisewhich varies as a function of coherence length, the phase noise having apeak value at a predetermined value of coherence length, wherein thecoherence length of the laser source is less than the predeterminedvalue. In certain embodiments, the sensor has a phase noise which variesas a function of coherence length, wherein the coherence length resultsin a phase noise less than about 2 μrad/√Hz, less than about 1 μrad/√Hz,or less than about 0.5 μrad/√Hz.

In certain embodiments, the first signal and the second signal have thesame frequency as the light from the laser source. In certainembodiments, the laser source has a mean wavelength stability greaterthan 1 part per million. In certain embodiments, the sensor is afiber-optic gyroscope comprising a standard Sagnac loop which comprisesthe coil.

In certain embodiments, a method of operating a fiber-optic sensor isprovided. The method comprises providing a fiber-optic sensor comprisingan optical fiber coil having a length and a laser source opticallycoupled to the coil. The laser source has a coherence length such thatthe sensor has a phase noise below a predetermined value. The coherencelength is less than the length. The method further comprises stabilizinga DC offset of the sensor. The method further comprises transmittinglight from the source to the coil as a first signal and a second signal.The first signal propagates along the coil in a first direction and thesecond signal propagates along the coil in a second direction oppositeto the first direction. The optical paths of the first signal and thesecond signal are substantially reciprocal with one another. The methodfurther comprises combining the first signal and the second signaltogether to generate a third signal.

In certain embodiments, a ratio of the coherence length to the length ofthe coil is greater than 0.1. In certain embodiments, the coherencelength is greater than 1 meter or is in a range between 200 microns and10 centimeters.

In certain embodiments, a method of configuring a fiber-optic sensor isprovided. The method comprises providing a fiber-optic sensor comprisinga laser source and an optical fiber coil having a length. The coil isoptically coupled to the laser source such that light from the lasersource is transmitted to the coil as a first signal propagating alongthe coil in a first direction and a second signal propagating along thecoil in a second direction opposite to the first direction. The opticalpaths of the first signal and the second signal are substantiallyreciprocal with one another and the first signal and the second signalare combined together after propagating through the coil to generate athird signal. The sensor has a phase noise which varies as a function ofcoherence length of the laser source. The method further comprisesselecting the laser source to have a coherence length such that thesensor has a phase noise less than a phase noise resulting when thesensor is driven by a broadband source.

In certain embodiments, the coherence length is less than the length. Incertain embodiments, a ratio of the coherence length to the length ofthe coil is greater than 0.1. In certain embodiments, the coherencelength is greater than 1 meter or is in a range between 200 microns and10 centimeters. In certain embodiments, the predetermined value is lessthan about 2 μrad/√Hz, less than about 1 μrad/√Hz, or less than about0.5 μrad/√Hz. In certain embodiments, the first signal and the secondsignal have the same frequency as the light from the laser source.

BRIEF DESCRIPTION OF THE DRAWINGS

FIGS. 1A-1E schematically illustrate example fiber-optic sensors inaccordance with certain embodiments described herein.

FIG. 2 shows a diagram of an example configuration of a fiber opticgyroscope (FOG) in accordance with certain embodiments described herein.

FIG. 3 shows a diagram of the four components of the output field fromthe fiber loop at port 1.

FIG. 4 shows simulated and experimental results of FOG noise vs. sourcecoherence length for a solid-core FOG and an air-core PBF FOG inaccordance with certain embodiments described herein.

FIG. 5 shows simulated results of FOG mean phase error vs. sourcecoherence length for the solid-core FOG and the air-core PBF FOG in FIG.4 using the numerical model described herein.

DETAILED DESCRIPTION

Simulations and experiment show that when the coherence length of alaser exceeds the length of the fiber loop in a fiber optic gyroscope(FOG), the noise due to coherent backscattering decreases. When thecoherence length is long enough, a gyroscope in accordance with certainembodiments described herein can have substantially the same sensitivityas the same gyroscope driven with a conventional broadband source. Oneexample of a laser source for certain embodiments described herein is asemiconductor laser. A particularly good wavelength range for the laseris around 1.5 μm, because it coincides with the lowest loss of thesilica-based fibers used in the gyroscope's sensing coil, but lasersources with other wavelengths are also compatible with certainembodiments described herein. A distributed-feedback (DFB) laser is agood candidate, although any semiconductor laser with an appropriatelynarrow linewidth and a stable enough frequency is applicable in certainembodiments. The InGaAsP semiconductor lasers around 1.5 μm developedfor the optical telecom industry are excellent candidates, because inorder to meet the telecom requirements they have been engineered to beultra-stable and single-frequency, as well as mechanically robust.Furthermore, by applying other noise reduction schemes, for examplemodulation of the laser frequency (a scheme described in U.S. patentapplication Ser. No. 12/271,760, which is incorporated in its entiretyby reference herein), it is possible in certain embodiments to reducethe coherent backscattering noise below that of a conventional FOGdriven with a broadband source. The noise of a gyroscope in accordancewith certain embodiments described herein is then limited by shot noise.

This scheme has several advantages. First, a laser has a much morestable frequency than does a broadband source. Hence, when using alaser, the scale factor stability of a gyroscope in accordance withcertain embodiments described herein is enhanced. This stability isparticularly important for certain inertial navigation applications of agyroscope, which require a high stability of the scale factor over manyhours. Second, for a given input electrical power, a semiconductor laserproduces a higher output power than does a broadband source.Consequently, for the same input power, a laser-driven gyroscope inaccordance with certain embodiments described herein has a higher outputpower at the detector (which means lower detection noise).Alternatively, for the same detected power, a gyroscope in accordancewith certain embodiments described herein has a lower electrical powerrequirement. Either case is desirable in certain embodiments. Third,because the signal-to-noise ratio of a shot-noise-limited signalincreases as the power increases, the total noise of a laser-drivengyroscope in accordance with certain embodiments described herein can bereduced significantly by increasing the detected power, e.g., byincreasing the laser power. This means that the sensitivity to rotationof a gyroscope in accordance with certain embodiments described hereincan be increased by increasing the laser power. This is not true in aconventional gyroscope driven by a broadband source, because the noisein the broadband detected signal is ultimately (e.g., at high enoughdetected power) limited by excess noise associated with the broadbandnature of the light, and in this case it is well known that the angularrandom walk in the detected signal is independent of detected power,e.g., it does not improve with increased detected power.

The fact that the noise due to coherent backscattering deceases when thecoherence length of the source exceeds the fiber loop length has beenreported in M. Digonnet, S. Lloyd, and S. Fan, “Coherent BackscatteringNoise in Photonic-Bandgap Fiber Optic Gyroscope,” InternationalConference on Optical Fiber Sensors, Edinburgh, Scotland (October 2009),which is incorporated in its entirety by reference herein. Since theearly years of the gyroscope, when the deleterious effects of coherentbackscattering were first demonstrated, it has been assumed that as thecoherence length of a laser approaches the length of the loop, the noiselevel reaches a maximum, and further increasing the coherence lengthbeyond the loop length does not modify it. The rationale was that whenthe coherence length equals the loop length, all the scatterers presentin the sensing fiber contribute to noise, and the noise could not worsenor improve by increasing the coherence length. The argument wasfallacious because it fails to take into account the noise property ofthe source. As the coherence length of a laser increases, its phasenoise also decreases. This phase noise is at the root of the coherentbackscattering noise: it is the randomness in the phase of the photonsincident on the backscatterers that spell out the magnitude of thenoise, not some random phase acquired by backscattered photons at thebackscatterers. In work by our research group, this point was firstdemonstrated through numerical simulations and has subsequently beendemonstrated experimentally in a fiber optic gyroscope utilizing aconventional index-guiding fiber. This principle is, however, verybroad. It is applicable for any kind of fiber, includingphotonic-bandgap fibers (PBF), microstructured fibers with a solid coreand holey cladding, etc. It is also applicable for other forms ofwaveguides besides fiber.

As mentioned above, certain embodiments described herein can utilize aPBF. The advantages of using a PBF in the sensing loop of a FOG can befound for example in H. K. Kim, M. J. F. Digonnet, and G. S. Kino,“Air-core photonic-bandgap fiber gyroscope,” J. Lightwave Techn., Vol.24, No. 8, 3169-3174 (August 2006). This fundamental change has severalmajor benefits, all of which stem from the fact that in a PBF, lighttravels mostly in air. First, air has a much weaker Kerr constant thansilica, so the Kerr-induced drift should be very small, even if a laseris used as a light source. Second, because air scatters much less thansilica, coherent-scattering noise in a PBF should ultimately bemanageable, provided backscattering due to random variations in thefiber cross section can be kept to a low enough level. As a result ofthese two improvements, the use of a laser instead of an SFS is anappealing new possibility in an air-core FOG. Reverting to a laser hasseveral key advantages, as explained above, including elimination ofexcess noise and thus increased sensitivity to rotation, far greatermean-wavelength stability, and reduced electrical power consumption andsize. Third, air has a much weaker refractive index dependence ontemperature than silica. Consequently, the well-known and troublesomelong-term drift due to asymmetric, time-varying temperature gradientsalong the sensing coil (known as the Shupe effect) is reduced. This isimportant because even though the Shupe effect is mitigated in practicalgyroscopes by special windings (e.g., quadrupolar) of the fiber loop, itis not completely gone, and the residual drift is often too large forhigh-end applications. Fourth, the Faraday constant is much weaker inair than in silica. The signal drift induced by the Earth's magneticfield acting on the coil is therefore significantly reduced, as can beinferred from a measurement of the Verdet constant of an air-core fiber(see H. Wen, M. A. Terrel, H. K. Kim, M. J. F. Digonnet, S. Fan, and G.S. Kino, “Measurements of the birefringence and Verdet constant in anair-core fiber,” IEEE J. of Lightwave Technol. Vol. 27, No. 15,3194-3201 (August 2009)). This should reduce the amount of mu-metalshielding of the fiber coil, which results in a cost, space, and weightsaving.

Several of these improvements have been demonstrated experimentally. Ina PBF FOG prototype operated with a narrowband laser, our research groupmeasured a 6.5-fold reduction in thermal drift, in agreement withtheory. See S. Blin, H. K. Kim, M. J. F. Digonnet, and G. S. Kino,“Reduced thermal sensitivity of a fiber-optic gyroscope using anair-core photonic-bandgap fiber,” J. Lightwave Techn., Vol. 25, No. 3,861-865 (March 2007). With straightforward fiber design improvements,this figure could be increased to a factor of ˜23. In the samegyroscope, a 250-fold reduction in the effective Kerr constant over aconventional FOG (4.4×10⁻³ vs. 1.1 rad/W/km) was also measured, in goodaccord with theory. See V. Dangui, M. J. F. Digonnet, and G. S. Kino,“Laser-driven photonic-bandgap fiber optic gyroscope with negligibleKerr-induced drift,” Opt. Lett., Vol. 34, No. 7, 875-877 (April 2009).This has led to the demonstration of a laser-driven air-core FOG with aninferred Kerr-induced long-term drift low enough to meet the requirementfor a 10-h flight. A reduction in the Faraday constant of a factor of 90compared to a conventional fiber was also measured. See H. Wen, M.Terrel, M. J. F. Digonnet, and S. Fan, cited previously.

FIG. 1A schematically illustrates an example fiber-optic sensor 10 inaccordance with certain embodiments described herein. The sensor 10comprises an optical fiber coil 20 and at least one optical coupler 30optically coupled to the coil 20. The sensor 10 further comprises alaser source 40 optically coupled to the at least one optical coupler30. Light from the laser source 40 is transmitted by the at least oneoptical coupler 30 to the coil 20 as a first signal 52 propagating alongthe coil 20 in a first direction 54 and a second signal 56 propagatingalong the coil 20 in a second direction 58 opposite to the firstdirection 54. The optical paths of the first signal 52 and the secondsignal 56 are substantially reciprocal with one another and the firstsignal 52 and the second signal 56 are combined together by the at leastone optical coupler 30 to generate a third signal 60. In certainembodiments, the laser source 40 has a coherence length in accordancewith the discussion below.

In certain embodiments, the fiber-optic sensor 10 is a Sagnac-basedfiber-optic sensor, as schematically illustrated by FIG. 1A. The sensor10 of certain embodiments is a FOG that is sensitive to rotations of thecoil 20 (e.g., the power carried by the third signal 60 changes as therate of rotation (e.g., in degrees per hour) applied to the coil 20varies). In certain other embodiments, the sensor 10 is configured to besensitive to one or more other perturbations, including but not limitedto, acoustic, thermal, and magnetic perturbations. The sensor 10 ofcertain embodiments thereby provides for detection of one or more of thefollowing: rotational movements, acoustic fields, thermal transients,and magnetic fields. The sensor 10 of certain embodiments is configuredto be used for one or more purposes, including but not limited to, as acompass, as a gyrocompass, and as a motion sensor. Persons skilled inthe art will recognize that while the majority of the discussion belowis presented with regard to FOGs, other fiber-optic sensors are alsocompatible with certain embodiments described herein.

The coil 20 of certain embodiments comprises a plurality ofsubstantially concentric loops. In certain embodiments, the coil 20comprises a conventional optical fiber (e.g., a single-mode fiber suchas the SMF-28® optical fiber available from Corning, Inc. of Corning,N.Y.). In certain other embodiments, the coil 20 comprises an air-coreoptical fiber (e.g., a hollow-core photonic-bandgap fiber such as the7-cell HC-1550-02 optical fiber available from Crystal Fibre A/S ofBirkerød, Denmark). In certain embodiments, the air-core optical fiberadvantageously provides a reduction of one or more of the Kerr effect,the Faraday effect, and the Shupe (thermal) effect, as compared toconventional optical fibers. See, e.g., U.S. Pat. Appl. Publ. No.2008/0030741 A1 and H. K. Kim, V. Dangui, M. Digonnet, and G. Kino,“Fiber-optic gyroscope using an air-core photonic-bandgap fiber,”Proceedings of the SPIE, vol. 5855, no. 1, pp. 198-201 (2005), each ofwhich is incorporated in its entirety by reference herein. However, thebackscattering coefficient of existing air-core optical fibers isactually higher than that in conventional solid-core optical fibers (byup to about one order of magnitude), thereby severely limiting thesensitivity of a laser-driven air-core fiber-optic sensor (e.g., FOG).

If the Kerr effect is still too large and thus introduces a detrimentalphase drift that degrades the performance of the fiber optical system,other methods can also be employed to reduce the Kerr effect in a fiberoptic system implemented in a Sagnac interferometer including anarrowband source comprising a light-emitting device in combination withan amplitude modulator. The optical signal from the light-emittingdevice is modulated by the amplitude modulator. In certain embodiments,the amplitude modulator produces a square-wave modulation, and incertain embodiments, the resulting light output from the narrowbandsource has a modulation duty cycle of about 50%. The modulation ismaintained in certain embodiments at a sufficiently stable duty cycle.As discussed, for example, in U.S. Pat. No. 4,773,759, and in R. A.Bergh et al., Compensation of the Optical Kerr Effect in Fiber-OpticGyroscopes, Optics Letters, Vol. 7, 1992, pages 282-284, suchsquare-wave modulation effectively cancels the Kerr effect in afiber-optic gyroscope.

However, with straightforward technical improvements, air-core opticalfibers can have a dramatically reduced backscattering level which ismuch lower than prevails in current air-core fibers. For example, onemethod for reducing backscattering of an air-core optical fiber is toincrease the diameter of the fiber core, e.g., by removing 19 tubes fromthe fiber preform to form the core, rather than 7 as is done for mostcurrent air-core optical fibers. A second method includes designing thefiber such that it has a wider bandgap. This can be accomplished, forexample, by increasing the fiber's air-filling ratio. A third approachfor reducing the level of backscattering is to increase the speed atwhich the fibers are drawn, which in itself requires adjusting otherfabrication and preform parameters, such as the temperature of the meltzone, the pressure of the gas applied to the preform's tubes, theviscosity and/or composition of the glass, etc. These methods ofreducing backscattering in an air-core optical fiber, and their physicalorigin and mathematical justifications (in some cases), can be found inVinayak Dangui's Doctorate Thesis, Laser-Driven Air-CorePhotonic-Bandgap Fiber Optic Gyroscope, Electrical EngineeringDepartment, Stanford University, October 2007, in particular in Section5.3.7, which is incorporated in its entirety by reference herein. Otheroptical fibers are also compatible with various embodiments describedherein.

In certain embodiments, as schematically illustrated by FIG. 1A, the atleast one optical coupler 30 comprises a first optical coupler 70comprising a first port 72, a second port 74, and a third port 76. Forexample, the first optical coupler 70 can comprise a 3-dB opticalcoupler, as schematically illustrated in FIG. 1A. The first opticalcoupler 70 of certain embodiments comprises additional ports. In certainembodiments, the second port 74 is optically coupled to a first end 22of the coil 20 and the third port 76 is optically coupled to a secondend 24 of the coil 20, as schematically illustrated by FIG. 1A. Lightgenerated by the laser source 40 received at the first port 72 is splitinto the first signal 52 and the second signal 56. The first signal 52is transmitted by the second port 74 to the first end 22 of the coil 20to propagate in the first direction 54 (e.g., clockwise) along the coil20, and is transmitted by the second end 24 of the coil 20 and the thirdport 76 to the first port 72. The second signal 56 is transmitted by thethird port 76 to the second end 24 of the coil 20 to propagate in thesecond direction 58 (e.g., counterclockwise) along the coil 20, and istransmitted by the first end 22 of the coil 20 and the second port 74 tothe first port 72. Thus, the first signal 52 and the second signal 56counterpropagate through the coil 20 and are recombined together by thefirst optical coupler 70.

In such a configuration, the optical paths of the first signal 52 andthe second signal 56 are substantially reciprocal with one another. Theterm “reciprocal” as used herein includes its broadest reasonableinterpretation, including, but not limited to, optical paths which havesubstantially the same optical length and which have substantially equalresponses to perturbations (e.g., thermal variations). For example, forlight traveling from a first state (“state” including polarizationstate, phase, but not amplitude) at point A to a second state at pointB, light propagation is reciprocal if upon reversing the direction oflight at point B, the light (now starting in the second state at pointB) gets back to point A again in the first state. For certainembodiments described herein, because the two signals 52, 56 travelalong the same optical path, their propagation is basically orsubstantially reciprocal such that the phase accumulated by the firstsignal 52 as it travels around the entire coil 20 in one direction isequal to the phase accumulated by the second signal 56 as it travelsaround the entire coil 20 in the opposite direction. This reciprocitywould be absolute in the absence of nature's very few non-reciprocaleffects, such as the Faraday effect (resulting from exposure to amagnetic field) and the Sagnac effect (resulting from exposure to arotation), and in the absence of asymmetric time-dependent effects (suchas dynamic perturbations, e.g., pressure or temperature variations),applied asymmetrically to any fraction or all of the sensing coil 20.However, this reciprocity is not absolute unless nonreciprocal effectsare all exactly zero, which means, in particular, that the two signals52, 56 must be in the same state of polarization (SOP) at every pointalong the coil 20 (although the SOP of each signal does not have to bethe same at every point along the coil 20). In this context, the term“substantially reciprocal” recognizes that canceling these residualnon-reciprocal effects is never complete. Examples of systems comprisingsubstantially reciprocal optical paths include, but are not limited to,common-path interferometers and common-mode interferometers. Examples ofnon-reciprocal optical paths are found in J. Zheng, “All-fibersingle-mode fiber frequency-modulated continuous-wave Sagnac gyroscope,”Optics Letters, Vol. 30, pp. 17-19 (2005) which discloses an unbalancedinterferometer.

In certain embodiments, as schematically illustrated by FIG. 1A, the atleast one optical coupler 30 further comprises a second optical coupler80 comprising a first port 82, a second port 84, and a third port 86.The second optical coupler 80 of certain embodiments comprisesadditional ports. For example, the second optical coupler 80 cancomprise an optical circulator, as schematically illustrated by FIG. 1A.In certain embodiments, the first port 82 receives light generated bythe laser source 40 (e.g., the first port 82 is optically coupled to thelaser source 40), the second port 84 is optically coupled to the firstport 72 of the first optical coupler 70, and the third port 86 isoptically coupled to a detection system 90. Light received by the firstport 82 from the laser source 40 is transmitted through the second port84 to the first port 72 of the first optical coupler 70. Light (e.g.,the third signal 60) received by the second port 84 from the first port72 of the first optical coupler 70 is transmitted through the third port86 to the detection system 90. Other configurations of the at least oneoptical coupler 30 are also compatible with certain embodimentsdescribed herein. For example, the at least one optical coupler 30 cancomprise additional or fewer optical elements, and the second opticalcoupler 80 can comprise a 3-dB optical coupler. As described more fullybelow, in certain embodiments, the sensor 10 can comprise a polarizerwhich can be used advantageously to achieve polarization reciprocity.

When the coil 20 is not rotated, the first signal 52 and the secondsignal 56 returning to the first port 72 after propagating through thecommon-path interferometer formed by the coil 20 and the first coupler70 are recombined in phase. If a dynamic perturbation is applied to thecoil 20 anywhere but in the mid-point of the coil 20 (identified by asmall cross on the coil 20 of FIG. 1A), the counterpropagating firstsignal 52 and second signal 56 experience a phase differential. When thetwo signals 52, 56 are recombined by the at least one optical coupler 30at the port 72, this phase differential results in an amplitudedifferential in the third signal 60 at the port 72, which is detected bythe detector system 90. This amplitude differential contains theinformation about the perturbation. A rotation of the coil 20 alsoinduces a phase shift whose amplitude is proportional to the rotationrate. When the sensor 10 is not perturbed (e.g., when the FOG is withoutrotation), the signal returning from an ideal FOG contains spectralcomponents at even multiples of the modulation frequency (dc included),but does not return any signal at f₀. However, any perturbation,including backscattering noise, will induce a component at f₀. Thus, thesignal of interest in certain embodiments is modulated at f₀.

In certain embodiments, the laser source 40 has a mean wavelength in arange between about 1.48 μm and about 1.6 μm (e.g., about 1.5 μm), whileother wavelengths are also compatible with certain embodiments describedherein. The mean wavelength of the laser source 40 of certainembodiments is stable to within about one part per million or better.The greater stability of the mean wavelength of certain embodiments, ascompared to an SFS, advantageously provides a greater scale-factorstability for the FOG.

In certain embodiments, as described more fully below, the laser source40 comprises a laser having a narrow bandwidth such that its coherencelength is greater than the optical length of the coil 20 (e.g., in arange from hundreds of meters to thousands of kilometers), equal to theoptical length of the coil 20 (e.g., a few hundred meters or longer), orless than the optical length of the coil 20 but considerably longer thanthe coherence length of a broadband source (which for a typical SFS istens of microns).

The term “coherence length” as used herein has its broadest reasonableinterpretation, including but not limited to, the coherence length inthe material in which the light substantially propagates within thefiber coil. The term “ratio of coherence length to length of the coil”as used herein has its broadest reasonable interpretation, including butnot limited to, the coherence length in the material in which the lightsubstantially propagates within the fiber coil divided by the physicallength of the fiber of the coil. For example, in the context of anair-core fiber, unless otherwise specified, the coherence length isdefined as the coherence length measured in air, and the ratio ofcoherence length to length of the coil is defined as the coherencelength measured in air divided by the physical length of the fiber ofthe coil. For another example, in the context of a conventionalsolid-core fiber, unless otherwise specified, the coherence length isdefined as the coherence length measured in the material of which thefiber is made (e.g., silica for a conventional solid-core fiber), andthe ratio of coherence length to length of the coil is defined as thecoherence length measured in the material of which the fiber is made(e.g., silica) divided by the physical length of the fiber of the coil.

In certain embodiments in which the coherence length is greater than orequal to the length of the coil 20, the ratio of the coherence length tothe length of the coil 20 is greater than 1, greater than 1.1, greaterthan 1.5, greater than 2, greater than 5, greater than 10, greater than100, or greater than 1000. In principle, the coherence length of thesource in certain embodiments can be selected to be as long as possibleto reduce the backscattering noise as much as possible. For example,referring to FIG. 4, described more fully below, increasing thecoherence length beyond the value corresponding to the peak or maximumcoherent backscattering noise will result in a reduction of the coherentbackscattering noise from its peak value. In certain embodiments, thereduction is roughly linear on a log-log scale, and the net noisereduction can be characterized by a factor of ten every time thecoherence length is increased by a factor of 200. In certainembodiments, however, it can be sufficient to reduce the backscatteringnoise to just below the next dominant source of noise, for example shotnoise.

In certain embodiments in which the coherence length is less than orequal to the length of the coil 20, the ratio of the coherence length tothe length of the coil 20 is greater than 0.00001, greater than 0.00005,greater than 0.0001, greater than 0.0005, greater than 0.001, greaterthan 0.005, greater than 0.01, greater than 0.05, greater than 0.1, in arange between 0.1 and 1, in a range between 0.3 and 1, or in a rangebetween 0.5 and 1. In certain embodiments, the ratio of the coherencelength to the length of the coil 20 is in a range between 0.5 and 1.5.

In certain embodiments, the coherence length is in a range between 100microns and 10 centimeters, in a range between 100 microns and 5centimeters, in a range between 100 microns and 1 centimeter, in a rangebetween 200 microns and 10 centimeters, in a range between 200 micronsand 5 centimeters, in a range between 200 microns and 1 centimeter, in arange between 500 microns and 10 centimeters, in a range between 500microns and 5 centimeters, in a range between 500 microns and 1centimeter, in a range between 1 millimeter and 10 centimeters, in arange between 1 millimeter and 5 centimeters, in a range between 1millimeter and 1 centimeter, in a range between 1 centimeter and 10centimeters, in a range between 1 centimeter and 5 centimeters, greaterthan 1 meter, in a range between 1 meter and 1 kilometer, in a rangebetween 1 meter and 500 meters, in a range between 1 meter and 100meters, in a range between 10 meters and 100 meters, in a range between10 meters and 500 meters, in a range between 10 meters and 1 kilometer,in a range between 100 meters and 500 meters, in a range between 100meters and 1 kilometer, in a range between 100 meters and 10 kilometers,in a range between 1 kilometer and 10 kilometers, in a range between 10kilometers and 100 kilometers, or in a range between 1 kilometer and 100kilometers. In certain such embodiments, the coherence length is lessthan or equal to the length of the coil 20, while in certain otherembodiments, the coherence length is greater than the length of the coil20. The coherence length is selected in certain such embodiments toprovide a noise level below a desired level, and the selection of thecoherence length is based on calculations as described more fully below.

In certain embodiments, the bandwidth of the laser source 40 issufficiently narrow such that the sensor 10 is substantially free fromexcess noise due to beating between the spectral components of the lasersource 40 (e.g., the excess noise is below the shot noise of thedetected signal). Examples of lasers compatible with certain embodimentsdescribed herein include, but are not limited to, external-cavitysemiconductor diode lasers and distributed feedback fiber lasers. Incertain embodiments, the distributed-feedback fiber laser is moresuitable since it is more compact and robust than an external-cavitysemiconductor diode laser. In certain embodiments, the laser frequencyis modulated in some pattern (e.g., sinusoidal, saw-tooth, etc.) at aselected frequency f_(m), as described in U.S. patent application Ser.No. 12/271,760, entitled “Low-Noise Fiber-Optic Sensor Utilizing a LaserSource,” filed on Nov. 14, 2008, and incorporated in its entirety byreference herein. In certain such embodiments, the frequency modulationis selected to provide a reduction of the excess noise (and thusimproved sensitivity, e.g. to rotation for a FOG) and in certainembodiments, to provide a reduction of the backscattered noise.

Coherent backscattering due to the interaction between light andinhomogeneities in the local index of refraction of a medium is known tobe a primary noise source in a variety of Sagnac interferometer-basedsensors such as fiber optic gyroscopes, acoustic sensors, etc. Whenlight encounters such a local inhomogeneity, it is scattered in variousdirections. The portion of the scattered light in the reverse directionthat is within the acceptance cone of the fiber will couple into thereverse propagating mode. Upon exiting the coil, this light willinterfere with each of the primary waves, producing an error signal. Theoptical paths of the scattered light and the primary light are no longerreciprocal, so that local variations in the fiber propagation constantdue to temperature transients or fluctuating magnetic fields, as well asphase fluctuations in the source will cause the error signal due tobackscattering to fluctuate in time when the interference that occurs iscoherent. The root mean square (RMS) fluctuations in this error signallimit the minimum sensitivity of Sagnac-loop-based sensors such as theFOG. In the case of the FOG, this type of noise is often characterizedby the FOG random walk, given in units of deg/√hr.

FIG. 1B schematically illustrates a single scatterer S at a position zin the optical fiber coil 20 of a Sagnac fiber-optic sensor 100 inaccordance with certain embodiments described herein. The coil 20schematically illustrated by FIG. 1B includes a phase modulator 130, asdescribed more fully below. The sensor 100 also comprises a detector 90comprising a photodiode 92 and detector circuitry 94. Person skilled inthe art are able to provide a detector 90 compatible with certainembodiments described herein. In certain embodiments, the phasemodulator 130 biases the interferometer in quadrature, as described inH. C. Lefevre, “The Fiber-Optic Gyroscope,” Artech House, Inc., Norwood,Mass. (1993). The sensor 100 of FIG. 1B is an example of a fiber-opticgyroscope comprising a standard Sagnac loop, which comprises a coil 20closed upon itself by an optical coupler, e.g., a 3-dB fiber coupler. Incertain embodiments, the period of the phase modulation by the phasemodulator 130 is twice the time-of-flight in the coil 20, and thefrequency of this phase modulation is referred to as the properfrequency f₀ of the sensor 100. In certain embodiments, the modulationfrequency of the phase modulator 130 is equal to the proper frequency f₀of the coil 20. This selection of frequency has a number of advantageousbenefits, including maximizing the sensitivity of the FOG to rotation,as described by H. C. Lefèvre, cited above. Another beneficial effect ofthis phase modulation is that when the coil 20 is rotated, theinterference signal caused by this rotation at the output of the coil 20is centered at frequency f₀.

Backscattering noise arises from the interaction at the detector of thefirst signal 52 and the generally weaker signal generated bybackscattering of the second signal 56 off scatterers (e.g., thescatterer S at position z). The small amount of backscattered lighttravels back to the at least one optical coupler 30, where it interfereswith the first signal 52, thus generating noise on the first signal 52(due to the random character of both the phase of the photons in thefirst signal 52 and the phase and amplitude of the reflection off thescatterer). Since in this direction, by the time they interact both thefirst signal 52 and the backscattered signal have traveled through thephase modulator 130, the spurious signal resulting from theirinterference occurs at frequency f₀. Since the rotation-induced signalon the FOG output signal also occurs at f₀ (see, H. C. Lefèvre, citedabove), this spurious signal is indistinguishable from the rotationsignal of the FOG, and it therefore constitutes a source of error. Inthe opposite direction, the main difference, in the example sensor 100of FIG. 1B, is that by the time the second signal 56 and thebackscattered signal due to backscattering of the first signal 52 offscatterers interact, only the second signal 56 has traveled through thephase modulator 130. The reason is two fold. First, the backscatteredsignal was generated from the first signal 52 backscattering from thescatterer at position z, which occurs at a time when the first signal 52had not yet traveled through the phase modulator 130 and thus had notyet been modulated. Second, because this particular backscattered signaltravels counterclockwise, it also never travels through the phasemodulator 130. As a result, in this particular configuration, the secondsignal 56 does not carry any coherent backscattering noise at f₀.

Because this interference process between main and backscattered signalsis coherent, only scatterers located along a segment of the coil 20centered on the coil's midpoint and along a length of the coil 20approximately equal to the coherence length of the source 40 contributeto the coherent backscattering. The scatterers located along the rest ofthe coil 20 produce a backscattered signal that is not temporallycoherent with the main signal, thereby producing intensity noise,instead of phase noise. This noise is considerably weaker than coherentbackscattering noise. In a Sagnac interferometer utilizing a broadbandsource, which has a short coherence length (typically tens of microns),the coherent backscattering noise is therefore very weak. As pointed outearlier, when such a source is used, the dominant noise of source istypically excess noise, not backscattering noise. On the other hand,utilizing a narrow-bandwidth laser source instead of a broadband sourcecan result in dramatically enhanced noise due to the greater portion ofthe optical fiber coil 20 that produces coherent backscattering noise,because the coherence length of the laser source (typically 1 cm orlonger, and usually much longer, up to thousands of km) is considerablylonger than that of a broadband source. The coherence length of thelaser source can be typically a fraction of the length of the opticalfiber coil 20 (e.g., 0.1% of the length of the coil 20, which can be afew hundred meters or longer) or longer. Therefore, all the scatterersalong the optical fiber coil 20 contribute to the coherentbackscattering noise. In certain embodiments, this backscattering noiseis advantageously reduced by sweeping or modulating the frequency of thelaser source 40 and filtering the detected signal, as described in U.S.patent application Ser. No. 12/271,760, entitled “Low-Noise Fiber-OpticSensor Utilizing a Laser Source,” filed on Nov. 14, 2008, andincorporated in its entirety by reference herein. The purpose of thepolarization controller 120 in certain embodiments is to control thebirefringence of the coil 20 and to ensure that the state ofpolarization of the signal output by the coil 20 is aligned with respectto the polarizer transmission axis. This has two effects: (i) itmaximizes the optical power transmitted by the polarizer 110 back to theoptical circulator 80 and the photodiode 92; and (ii) it ensures thatthe two signals counterpropagating through the coil 20 have the samestate of polarization (SOP) at every point (although again, the SOP ofeach signal may not be the same at every point along the loop). Asdiscussed above, this can advantageously be used to ensure substantialreciprocity of the Sagnac interferometer.

FIG. 1C schematically illustrates another example sensor 101 inaccordance with certain embodiments described herein. The sensor 101 ofFIG. 1C is another example of a fiber-optic gyroscope comprising astandard Sagnac loop which comprises the coil 20. The sensor 101 of FIG.1C can be a FOG in the minimum configuration (see, e.g., H. C. Lefèvre,“The Fiber-Optic Gyroscope,” Artech House, Inc., Norwood, Mass. (1993)).Light from the laser source 40 is transmitted to the second opticalcoupler 80 (e.g., an optical circulator), through a polarizer 110, tothe first optical coupler 70 which is closed upon itself by apolarization controller 120, the optical fiber coil 20, and anelectro-optic (EO) phase modulator 130. The phase modulator 130 can beused to bias the sensor 101 in quadrature, thus improving thesensitivity of the sensor 101. In certain embodiments, the polarizer 110and the phase modulator 130 are fiber-based or fiber-pigtailedcomponents which are commercially available from a number of vendors andmanufacturers (e.g., JDS Uniphase Corp. of Milpitas, Calif.).

FIG. 1D schematically illustrates an example sensor 102 in accordancewith certain embodiments described herein. The sensor 102 of FIG. 1D isanother example of a fiber-optic gyroscope comprising a standard Sagnacloop which comprises the coil 20. In certain embodiments, the sensor 102comprises a polarization-maintaining (PM) fiber downstream from thepolarizer 110 (e.g., in the coil 20, between the polarizer 110 and thefirst optical coupler 70, and/or within the first optical coupler 70).In certain such embodiments, the entire optical path downstream from thepolarizer 110 is PM fiber. In certain embodiments, the sensor 102utilizes PM fiber throughout (i.e., downstream from the source 40). Byutilizing PM fiber either along the entire optical path downstream fromthe polarizer 110 or throughout the sensor 102, certain embodimentsobviate the use of the polarization controller 120 of the sensor 101 ofFIG. 1C. Certain such embodiments advantageously avoid the need toadjust the polarization controller 120 (either manually, which cannot bedone for an actual FOG, or with complicated feedback systems, which addcost and complexity). In certain other embodiments, a polarizing fibercan be used instead of the polarization-maintaining fiber discussedabove. The phase modulator 130 of certain embodiments is driven by afunction generator 140 which is coupled to a lock-in amplifier 150 whichoutputs a signal to a computer system 160. The lock-in detection at theproper frequency of the sensor 101, 102 in certain embodiments canadvantageously improve the signal-to-noise ratio. With this phasemodulation, the returning signal of interest is centered at thefrequency of the phase modulation (i.e., at the proper frequency f₀).

In a manner similar to that discussed above with regard to the exampleconfiguration illustrated by FIG. 1B, for the sensor 101 schematicallyillustrated by FIG. 1C, the backscattered light due to only one of thecounterpropagating signals propagates through the phase modulator 130.For example, for the first signal 52 propagating through thepolarization controller 120 then through the rest of the coil 20 andthen through the phase modulator 130, any backscattered light producedwithin the coil 20 will propagate towards the polarization controller120 and away from the phase modulator 130 before reaching the firstoptical coupler 70. Conversely, for the second signal 56 propagatingthrough the phase modulator 130, then through the coil 20, and thenthrough the polarization controller 120, any backscattered lightproduced within the coil 20 will propagate through the phase modulator130 before reaching the first optical coupler 70. The backscatteredlight that does not propagate through the phase modulator 130 is thusnot phase modulated, and therefore does not contribute to thebackscattering noise at the detection frequency. Such a configuration isdifferent from other configurations (e.g., J. Zheng, “All-fibersingle-mode fiber frequency-modulated continuous-wave Sagnac gyroscope,”Optics Letters, Vol. 30, pp. 17-19 (2005) and J. Zheng, “Differentialbirefringent fiber frequency-modulated continuous-wave Sagnacgyroscope,” IEEE Photonics Technology Letters, Vol. 17, pp. 1498-1500(2005)) in which both backscattered signals are modulated so bothcontribute to the noise.

FIG. 1E schematically illustrates an example sensor 103 in accordancewith certain embodiments described herein. The sensor 103 of FIG. 1E isanother example of a fiber-optic gyroscope comprising a standard Sagnacloop which comprises the coil 20. The sensor 103 of FIG. 1E is anintegrated optic chip in which the FOG components are all made on a chip(LiNbO₃), in accordance with the standard method to make a commercialFOG. In certain such embodiments, the key components of the sensor 103,including but not limited to the first optical coupler 70 (e.g., a Yjunction), the second optical coupler 80 (e.g., a Y junction), thepolarizer 110, and the phase modulator 130, are all fabricated usingstandard technology on the same integrated optic chip, for example onLiNbO₃, which presents certain well-recognized advantages ofcompactness, mechanical stability, and ease and reduced cost oflarge-scale manufacturing. In certain embodiments, the coil 20 comprisesa polarization-maintaining fiber. In certain other embodiments, apolarization controller can be positioned at an appropriate point (e.g.,between the source 40 and the integrated optic chip) to control thepolarization of light entering the polarizer and to ensure that thesignal output state of polarization is aligned with respect to thepolarizer transmission axis, thereby maximizing the optical powertransmitted by the polarizer. In certain embodiments, the coil 20comprises a polarization-maintaining air-core fiber.

FIG. 2 is a diagram of an example experimental configuration of anopen-loop fiber gyroscope 200 in accordance with certain embodimentsdescribed herein. It uses a Sagnac interferometer made of a fiber coil20 comprising a plurality of substantially concentric loops 22 having alength of 235 m of either air-core fiber (e.g., HC-1550-02 from CrystalFibre) or conventional fiber (e.g., Corning's SMF-28 fiber), both coiledin a quadrupolar winding. The coil 20 further comprises a polarizationcontroller 120 and a fiber-pigtailed electro-optic phase modulator 130.A light source 40 comprises a laser 42, an optical isolator 44, and apolarization controller 46. Light from the light source 40 is sentthrough at least one optical coupler 30 (e.g., comprising a first 3-dBoptical coupler 70, a second 3-dB optical coupler 80, and a fiberpolarizer 110 as shown in FIG. 2) and coupled into the coil 20 in boththe clockwise (cw) and counterclockwise (ccw) directions. Aftertraveling around the coil 20, the signals return to and interfere at thefirst 3-dB optical coupler 70. The optical signal from the at least oneoptical coupler 30 (e.g., from the second 3-dB optical coupler 80) isreceived by the detector system 90 (e.g., comprising a photodiode 92, alock-in amplifier 150, and a computer 160, as schematically illustratedby FIG. 2). Through the Sagnac effect, this interference yields anoutput signal power that depends on the rotation rate imparted to thecoil 20. The interferometer was biased for maximum sensitivity with thefiber-pigtailed electro-optic phase modulator 26 placed asymmetricallyin the coil 20 and operated at the loop proper frequency (e.g., inresponse to signals from the function generator). See H. Lefèvre, citedpreviously, for details.

As discussed above, backscattering noise occurs when photons from the cwsignal are scattered into the ccw direction by Rayleigh scattering (andvice versa). These backscattered photons are captured by the fundamentalmode of the fiber, interfere with the primary ccw wave, and inducenoise. Scattered photons that are coherent with the primary wavesproduce interference and noise. Photons that are incoherent with theprimary waves produce intensity noise, which is typically negligible. Itfollows that coherent backscattering noise arises from the scattererslocated along a length of fiber equal to the coherence length of thelight L_(c) centered on the mid-point of the loop. When L_(c) is shortenough compared to the loop length L, as applies to a broadband source(L_(c)≈10-100 μm), coherent backscattering noise is negligible comparedto other sources of noise. When L_(c) is equal to a few meters orlonger, as applies to a laser, coherent noise typically dominates. Asdiscussed below, to predict theoretically the magnitude of this noise ina PBF FOG, the backscattering coefficient of an air-core fiber waspredicted, and backscattering noise in a Sagnac loop probed with a laserof arbitrary coherence was modeled, which had not been done previously.

Only one measurement of backscattering in an air-core fiber haspreviously been reported. See M. Wegmuller, M. Legré, N. Gisin, T. P.Hansen, C. Jakobsen, and J. Broeng, “Experimental investigation of thepolarization properties of a hollow core photonic bandgap fiber for 1550nm,” Opt. Expr., Vol. 13, No. 5, 1457-1467 (March 2005). It wasperformed in Crystal Fibers' AIR-10-1550 fiber at 1.55 μm. Based on aprivate communication with M. Legré, the backscattering coefficientinferred from this measurement was 1.58×10⁻⁶ m⁻¹. In a PBF, one expectsthat backscattering arises mainly from random perturbations of the fiberindex profile along the propagation direction, and that bulk scatteringin the silica portions of the fiber is negligible. As a confirmation, atheoretical model has been developed that calculates the mode couplingloss and backscattering coefficient of the fundamental (HE₁₁) mode of aPBF with random, azimuthally symmetric deformations of its indexprofile. See V. Dangui, M. J. F. Digonnet, and G. S. Kino, “Modeling ofthe propagation loss and backscattering in air-core photonic-bandgapfibers,” IEEE J. Lightwave Technol. Vol. 27, No. 17, 3783-3789 (Sep. 1,2009). When applied to the air-core fiber, this model predicted that toreplicate the fiber's measured loss (˜24 dB/km), for the range ofperturbation amplitude σ likely to prevail in this fiber (0.5 to 3% ofthe crystal period), the perturbation characteristic length D was in therange of ˜1 to ˜30 cm. That this range is much shorter than the typicalcorrelation length of a conventional single-mode fiber (˜3 m) isconsistent with the fact that PBFs are drawn at much slower speeds. Thissame perturbation yielded a predicted backscattering coefficient of1.5×10⁻⁶ m⁻¹, in good agreement with the experimental value. See, e.g.,M. Wegmuller, M. Legré, N. Gisin, T. P. Hansen, C. Jakobsen, and J.Broeng, cited above. The backscattering coefficient is thus ˜22 timeshigher in the 7-cell air-core fiber used in this work than in aconventional fiber such as Corning's SMF-28 fiber. Since thebackscattering noise in a FOG scales like the square root of thebackscattering coefficient, the backscattering noise in an air-core FOGis expected to be ˜4.7 times larger than in a conventional FOG.

Several reports have developed analytic methods and models forpredicting the effect of Rayleigh backscattering on the noise in a FOG.See e.g., K. Takada, “Calculation of Rayleigh backscattering noise infiber-optic gyroscopes,” J. Opt. Soc. Am. A, Vol. 2, No. 6 (June 1985);J. Mackintosh and B. Culshaw, “Analysis and observation of couplingratio dependence of Rayleigh backscattering noise in a fiber opticgyroscope,” J. Lightwave Tech., Vol. 7, No. 9, 1323-1328 (1989); and K.Krakenes and K Blotekjaer, “Effect of laser phase noise in Sagnacinterferometers,” J. Lightwave Tech., Vol. 11, No. 4, 643-653 (1993).However, in order to obtain a closed form solution, all these studiesrelied on the assumption that the coherence length L_(c) of the sourcewas much shorter than the loop length L. None of these studies examinedthe quantitative dependence of the coherent backscattering noise on thesource coherence length (i.e., they all specialized to an extremelyshort coherence length), and none incorporated the effect of phasemodulation. As explained above, a gyroscope in accordance with certainembodiments described herein can utilize a laser source with a coherencelength considerably longer than the typical coherence length of abroadband source, even up to coherence lengths on the order of, orlonger than, the loop length. In this configuration, the solutionspublished by prior studies fail to predict the proper noise due toRayleigh backscattering. As discussed herein, a new numerical model hasbeen developed to simulate this process in software and to predict thenoise in a gyroscope for an arbitrary coherence length. In addition, asdescribed herein, a new analytical model has been developed based on theequations published by Krakenes and Blotekjaer, but without theapproximation of negligible coherence length that they made towards thelater part of their derivation. However, a closed form analyticalsolution is obtained only in the special case of no phase modulation inthe interferometer. This exact solution was nevertheless useful inverifying that in the limit of no phase modulation, the numerical modeland the analytical model produced the same solutions, which they did.

The formulation described herein follows most closely that developed inKrakenes and Blotekjaer. Krakenes and Blotekjaer seem to go further intheir calculation before relying on the low coherence approximation thanother authors. Rather than the six-port coupler and DC phase bias theyexamined (see Krakenes and Blotekjaer), the configuration discussedherein (shown schematically in FIG. 3) is a more standard configurationof a sensor 300 (e.g., FOG) with a four-port coupler 30 and sinusoidalphase modulation for bias as an example of a certain embodimentdescribed herein.

As illustrated in FIG. 3, the output field from the fiber loop 20 atport 1 consists of four components, namely the two primary waves E₊ andE⁻ traveling in the clockwise and counterclockwise directions,respectively, and two scattered waves E₊ ^(b) and E⁻ ^(b). If thecomplex input field at port 1 is expressed as E₀e^(j[ω) ⁰ ^(·t+φ(t)]),where ω₀ is the center angular frequency of the source, and φ(t) is thesource phase noise for a finite source linewidth as a function of time,then the four components of the output field can be expressed as

$\begin{matrix}{{{E_{+}(t)} = {E_{0}a_{13}a_{14}{\mathbb{e}}^{t{\lbrack{{\omega_{0} \cdot {({t - {L\text{/}v}})}} + {\phi{({t - {L\text{/}v}})}} + {\Phi{(t)}} + {\Psi_{s}\text{/}2}}\rbrack}}{\mathbb{e}}^{{- \alpha}\; L\text{/}2}}}{{E_{-}(t)} = {E_{0}a_{14}a_{13}{\mathbb{e}}^{t{\lbrack{{\omega_{0} \cdot {({t - {L\text{/}v}})}} + {\phi{({t - {L\text{/}v}})}} + {\Phi{({t - {L\text{/}v}})}} - {\Psi_{s}\text{/}2}}\rbrack}}{\mathbb{e}}^{{- \alpha}\; L\text{/}2}}}{{E_{+}^{b}(t)} = {E_{0}a_{14}a_{14}{\int_{0}^{L}{{{jA}(z)}{\mathbb{e}}^{j{\lbrack{{\omega_{0} \cdot {({t - {2z\text{/}v}})}} + {\phi{({t - {2z\text{/}v}})}} + ~{(t)} + {\Phi{({t - {2z\text{/}v}})}}}\rbrack}}{\mathbb{e}}^{{- \alpha}\; z}\ {\mathbb{d}z}}}}}{{E_{-}^{b}(t)} = {E_{0}a_{13}a_{13}{\int_{0}^{L}{{{jA}( {L - \eta} )}{\mathbb{e}}^{j{\lbrack{{\omega_{0} \cdot {({t - {2\eta\text{/}v}})}} + {\phi{({t - {2\eta\text{/}v}})}}}\rbrack}}{\mathbb{e}}^{{- \alpha}\;\eta}\ {\mathbb{d}\eta}}}}}} & (1)\end{matrix}$

The coefficients a_(ij) represent the complex coupling coefficientsbetween ports n and m of the 2×2 coupler (the coupler is reciprocal, soa_(ji)=a_(ij)), v is the group velocity of the optical mode in thefiber, ψ_(s) is the rotation-induced Sagnac phase shift, α is theintensity attenuation coefficient of the fiber, A(z) is a randomvariable representing the Rayleigh scattering coefficient at position z,and Φ(t) is the loop phase modulation used for bias. The extra factor ofj in the expressions for E₊ ^(b) and E⁻ ^(b) is a consequence of theRayleigh scattering process, which adds a π/2 phase shift to the fieldthat is backscattered. See e.g., M. Nakazawa et. al, “Analyses ofoptical time-domain reflectometry for single-mode fibers and ofpolarization optical time-domain reflectometry forpolarization-maintaining fibers,” Optics Letters, Vol. 8, No. 2, 130-132(1983). A(z) is therefore assumed to be real.

At the output of the coupler 30 of the gyroscope, all of these wavesinterfere. The returning signal of the FOG is contained in theinterference of the primary waves E₊ and E⁻, while interferenceoccurring between a primary wave and a backscattered wave constitutes anerror signal. Random phase fluctuations present in any finite linewidthsource are converted by this interference process into intensityfluctuations in the output signal. These fluctuations ultimately limitthe minimum detectable rotation rate of the gyroscope. This source ofnoise is referred to as coherent backscattering noise, or Rayleighscattering noise.

A slightly more useful form of the fields represented in Eq. (1) can beobtained by defining{tilde over (E)}=a ₁₃ a ₁₄ E ₀ e ^(j[ω) ⁰ ^(·(t−L/v)+φ(t−L/v]) e^(−αL/2)  (2)Eq. (1) then becomes

$\begin{matrix}{{{E_{+}(t)} = {\overset{\sim}{E}{\mathbb{e}}^{j{\lbrack{{\Phi{(t)}} + {\Psi_{s}\text{/}2}}\rbrack}}}}{{E_{-}(t)} = {\overset{\sim}{E}{\mathbb{e}}^{j{\lbrack{{\Phi{({t - {L\text{/}v}})}} - {\Psi_{s}\text{/}2}}\rbrack}}}}{{E_{+}^{b}(t)} = {\overset{\sim}{E}\frac{a_{14}}{a_{13}}{\int_{0}^{L}{{{jA}(z)}{\mathbb{e}}^{j{\lbrack{{\omega_{0} \cdot \frac{L - {2z}}{v}} + {\overset{\sim}{\phi}{({{t - {L\text{/}v}},\frac{L - {2v}}{v}})}} + {\Phi{(t)}} + {\Phi{({t - {2z\text{/}v}})}}}\rbrack}}{\mathbb{e}}^{\frac{\alpha}{2}{({L - {2z}})}}\ {\mathbb{d}z}}}}}{{E_{-}^{b}(t)} = {\overset{\sim}{E}\frac{a_{13}}{a_{14}}{\int_{0}^{L}{{{jA}(\eta)}{\mathbb{e}}^{j{\lbrack{{\omega_{0} \cdot \frac{{2\eta} - L}{v}} + {\overset{\sim}{\phi}{({{t - {L\text{/}v}},\frac{{2\eta} - L}{v}})}}}\rbrack}}{\mathbb{e}}^{\frac{\alpha}{2}{({{2\eta} - L})}}\ {\mathbb{d}\eta}}}}}} & (3)\end{matrix}$where {tilde over (φ)}(t,τ)=φ(t+τ)−φ(t) and the variable of integrationin the second integral is shifted for clarity.

In order to calculate E₊ ^(b) and E⁻ ^(b), the integration in Eq. (3)can be simulated by dividing the fiber into a large number N of shortsegments of equal length, each of which contains in its center ascatterer. The amplitude of the field backscattered by the segment atlocation z along the Sagnac loop is the random variable A(z) definedearlier. Environmental perturbations of the fiber cause thisdistribution to vary. The time constant of these variations is muchlonger than the loop delay; consequently, these perturbations contributeto a drift in the FOG signal rather than a noise, and they can beignored in this model. On the other hand, the phase of the fieldbackscattered is equal to the phase of the incident light at z plus aphase shift on reflection of π/2. The incident light has phase noise,which depends on the linewidth of the source. It is therefore therandomness in the phase of the backscattered photons, not in theiramplitude, that is mainly responsible for coherent backscattering noise.This phase noise mainly comes directly from the incident light itself,not the scatterers.

At the start of a simulation, a particular random distribution A(z) isselected for the fiber, from z=0 to z=L. For the reason provided above,A(z) is taken to be time independent.

By discretizing the fiber into N segments and approximating A(z) asexplained, the integral in the expression for E₊ ^(b) found in Eq. (3)becomes

$\begin{matrix}{{\int_{0}^{L}{{{jA}(z)}\varepsilon^{j{\lbrack{{\omega_{0} \cdot \frac{L - {2z}}{v}} + {\overset{\sim}{\phi}{({{t - {L\text{/}v}},\frac{L - {2z}}{v}})}} + {\Phi{(t)}} + {\Phi{({t - {2z\text{/}v}})}}}\rbrack}}{\mathbb{e}}^{\frac{\alpha}{2}{({L - {2z}})}}\ {\mathbb{d}z}}} \approx {\sum\limits_{n = 0}^{N = {L\text{/}\Delta\; z}}\;{{{jA}(n)}{\mathbb{e}}^{j{\lbrack{{\omega_{0} \cdot \frac{L - {n\; 2\;\Delta\; z}}{v}} + {\overset{\sim}{\phi}{({{t - {L\text{/}v}},\frac{L - {n\; 2\;\Delta\; z}}{v}})}} + {\Phi{(t)}} + {\Phi{({t - {n\; 2\;\Delta\; z\text{/}v}})}}}\rbrack}}{\mathbb{e}}^{\frac{\alpha}{2}{({L - {2\; n\;\Delta\; z}})}}\Delta\; z}}} & (4)\end{matrix}$with a similar expression for E⁻ ^(b).

For a statistically homogeneous fiber medium and for the length scaleunder consideration here (Δz is assumed much larger than one opticalwavelength), the autocorrelation function of A(n) is given by E.Brinkmeyer, “Analysis of the backscattering method for single-modefibers,” J. Opt. Soc. Am. Vol. 70, 1010-1012 (1980) as<A(n)A(n′)>=α_(b)δ_(nn′)  (5)where δ_(nn′) is the Kronecker delta and <•> represents an ensembleaverage.

The source phase is assumed to follow a Wiener-Levy process. As such,the phase difference between any two points in time depends on thetemporal delay between these points. The phase noise is then describedby a Gaussian probability function, with a width proportional to thelinewidth of the source.

So, if {tilde over (φ)}(t,τ)=φ(t+τ)−φ(t) then {tilde over (φ)}(t,τ) isdescribed by the probability density function

$\begin{matrix}{{P( \overset{\sim}{\phi} )} = {\frac{1}{\sqrt{2{{\pi\sigma}^{2}(\tau)}}}{\mathbb{e}}^{{- {\overset{\sim}{\phi}}^{2}}\text{/}2\;{\sigma^{2}{(\tau)}}}}} & (6)\end{matrix}$withσ²(τ)=2πΔf|τ|  (7)where Δf is the full-width at half-maximum (FWHM) of the source.Furthermore, the phase changes over two different intervals {tilde over(φ)}₁(t,τ₁) and {tilde over (φ)}₂(t,τ₂) is taken to be statisticallyindependent for non-overlapping time intervals.

With these assumptions, there is enough information to simulate theeffect of backscattering in a FOG with an arbitrary source coherence.The model can be used to calculate the total field backscattered fromthe cw signal in the ccw direction (and vice versa) by summing thecontributions of all N scatterers, taking into account the phase delaydue to propagation to each scatterer and back, the fiber loss, and thephase statistics of the source. The two primary and two backscatteredfields can then be added at the output of the coupler to obtain thetotal signal returning from the loop. This coherent addition(interference) converts the random phase fluctuations in the source intointensity fluctuations in the output signal. The simulations can berepeated for several hundred integration periods, each time with a newinput signal phase distribution, after which the noise and angularrandom walk of the gyroscope are obtained.

To carry out the simulation, random backscattering coefficients for eachsegment n in the discretized fiber can be generated in software from thestatistics given by Eq. 5. Similarly, the random walk of the sourcephase noise can be generated according to Eq. 6 for subsequent timeintervals. The calculations described by Eq. 4 can then be carried outto obtain the time dependent forms of E_(±) ^(b) and E⁻ ^(b). The totalerror signal measured at the detector can be calculated usingI _(b)(t)=[E ₊ ^(b)(t)+E ⁻ ^(b)(t)]·[E ₊*(t)+E ⁻*(t)]+cc  (8)

When a sinusoidal phase modulation is applied to the FOG loop forbiasing purpose, the error signal can be further processed by extractingthe in-phase component of the signal at the first harmonic of themodulation frequency, as is done in practice in a FOG. See again H.Lefèvre, previously cited. The standard deviation of this signal,normalized by the square root of the measurement bandwidth, is the FOGangular random walk, which quantifies the rotation-rate error caused bycoherent backscattering noise. Since the result obtained by this processis specific to the particular distribution of backscatteringcoefficients (A(n)) chosen for the discretized fiber, the entire processcan be repeated with different distributions of A(n) and averaged over alarge number of distributions to accurately predict the expected noisedue to Rayleigh backscattering.

Because the mathematical model outlined above was kept as general aspossible while relying on the fewest number of assumptions, it is apowerful tool for exploring various aspects of a FOG. One interest wasunderstanding the effect of linewidth on the noise properties of theRayleigh backscattered signal. FIG. 4 shows the result of using thismodel to calculate the average noise due to Rayleigh backscattering fora varying source linewidth. This curve of the phase noise versuscoherence length was simulated for a FOG in accordance with certainembodiments described herein with a conventional solid-core fiber(Corning's SMF-28 fiber) loop length of 235 m, a loop loss of 0.2 dB/km,a laser wavelength in vacuum of 1.5 μm, and a sinusoidal phasemodulation at the proper frequency f₀=425 kHz. It was generated byrunning the simulator for a sufficient number of iterations (about 100random distributions of scatterers along the fiber length) to guaranteeconvergence, and averaging the curves. As the coherence length dropswell below the loop length, a diminishing fraction of the scattererscontribute to the coherent noise, and the noise decreases. This behaviorhas been the rationale behind using a broadband light source as theprimary means for reducing backscattering noise for many years. As thelinewidth of the source broadens, the fields scattered by each of theindividual scatterers no longer interfere coherently, which averages outthe effects of the source phase fluctuations over time and reduces thenoise in the output signal. A new implication of this curve is that itenables one to predict quantitatively for the first time the exactbackscattering noise for a gyroscope operated with a coherence lengthintermediate between that of a broadband source (typically tens ofmicrons) and the length of the sensing coil (typically 100 m or more).In particular, it gives the ability to select the coherence length toachieve a desired level of backscattering noise.

In addition, the right portion of the curve exhibits a trend that hasnever been reported previously: when L_(c) increases above L, the noiseactually decreases. By using a highly coherent source, the noise canactually be brought to the level observed in a typical FOG operated witha broadband source. Increasing L_(c) beyond L does not add any morescatterers. While increasing the source coherence length increases thecoherent scattering length, once the source coherence exceeds the lengthof the loop, there is no more length to add. The reason why the noiseactually decreases is that increasing the coherence length reduces thephase fluctuations of the light, and hence the fluctuations in thebackscattered light. The mean value of the backscattered signal, e.g.,its power, is not reduced, but its standard deviation, e.g., the noise,is reduced. This ultimately leads to a reduction of the noise due toRayleigh backscattering for highly coherent sources, as shown in FIG. 4.The noise level is higher with the PBF, approximately by the predictedfactor of about 4.7, because this fiber has a higher backscatteringcoefficient.

FIG. 4 shows that on the right side of the peak in the curve, for boththe conventional solid-core fiber and the PBF, the noise decreasesroughly as the square root of the coherence length as the coherencelength is increased. This effect is nonetheless significant: with asufficiently coherent source, the backscattering noise can be reduced bymore than one order of magnitude compared to its peak value. Inaccordance with certain embodiments described herein, this novel effectis applicable to a conventional FOG, which can be driven with ahigh-coherence source. In certain such embodiments, the Kerr-induceddrift is reduced by other means, for example by applying a 50%square-wave amplitude modulation to the input light, as discussed above.For example, in a FOG using a conventional solid-core fiber loop, asource linewidth of 6 kHz (L_(c)≈1.6×10⁴ m in air) can be used and thenoise can be brought down to the excess noise measured in this same FOGdriven with a typical SFS, around 1 μrad/√Hz. In such a FOG, because asolid-core fiber is used in the sensing coil, the Kerr-induced drift maybe too strong, but in certain embodiments, this deleterious effect canbe reduced by applying a 50% square-wave amplitude modulation to theinput laser light, in a manner that is well known in the art. Such ahigh-coherence source is readily available commercially.

In accordance with certain embodiments described herein, the air-coreFOG can use a high-coherence source and be less affected by such noisebecause the use of a laser results in a considerably weakernon-reciprocal Kerr-induced phase drift. For the 7-cell PBF inaccordance with certain embodiments described herein, the correspondinglinewidth to reduce the backscattering noise of an air-core FOG to theexcess noise of a conventional FOG using a solid-core fiber and abroadband source is about 320 Hz, which is currently difficult toachieve. Two straightforward steps can be used with larger linewidths incertain embodiments described herein. First, the fiber can be replacedby a 19-cell fiber, which has a much lower loss (˜1.2 dB/km) andtherefore expectedly a much lower backscattering coefficient. See B. J.Mangan, et al., “Low loss (1.7 dB/km) hollow core photonic bandgapfiber,” in Proc. Opt. Fiber. Commun. Conf. (2004), paper PDP24. Second,the frequency of the laser can be modulated to shift the coherentbackscattering interference energy away from the signal frequency. SeeU.S. patent application Ser. No. 12/271,760, filed Nov. 14, 2008 andincorporated in its entirety by reference herein; S. Blin, M. J. F.Digonnet, and G. S. Kino, “Fiber optic gyroscope operated with afrequency-modulated laser,” Conf. on Optical Fiber Sensors, Perth,Australia, Proc. SPIE Vol. 7004, 70044X-1-4 (April 2008). Combined, incertain embodiments, these two improvements can bring the backscatteringnoise below the excess-noise limit with a practical narrow linewidth.

Another aspect of certain embodiments described herein relates to theaverage backscattered power generated by Rayleigh backscattering in thesensing coil. As mentioned above, as the coherence length of the sourceis increased, the total number of scatterers involved in the coherentbackscattering process increases, and the mean power that isbackscattered by the fiber (in both direction) increases. Thisadditional signal interferes with the two primary waves at the loopcoupler, which has two effects. First, it produces a mean DC offset inthe output of the gyroscope. Second, because the primary waves and thebackscattered waves travel different optical paths (e.g., some of thescattered photons were generated near one end of the loop, and havetherefore experienced a different overall phase shift than the primarywaves), their relative phase varies with temperature. As a consequence,the coherent sum of these four fields depends on temperature. As thetemperature of the coil fiber varies, the total output of the gyrocopecoil varies, in a manner that is indistinguishable from a rotation.

The predicted magnitude of this effect can be seen in FIG. 5, whichplots the mean phase error (or offset) in the two modeled gyroscopes(FOG with solid-core SMF-28 fiber and FOG with air-core PBF) as afunction of source coherence length. For very short coherence length,for example available from an SFS, the coherence length in silica is ofthe order of 10 μm and the mean phase error in an SMF-28 FOG isnegligible (out of the range shown in the figure). With a laser ofcoherence length of ˜2 km or longer, the phase offset is approximatelymaximum and equal to ˜2 mrad. The offset is higher for the PBF FOG (seetop curve in FIG. 5) because again the backscattering is higher.

FIG. 5 demonstrates that when selecting the coherence length of thelaser in certain embodiments, the DC offset that will result from theuse of a laser is advantageously considered. For longer coherencelengths, the noise drops (see FIG. 4) but the DC offset (and thereforethe long-term drift) increases. For both fibers, this offset and theoffset drift associated with it are undesirable. However, if the offsetis stable, it can be measured while the FOG is at rest, then subtractedfrom the measured signal when the FOG is operating and rotated. If theFOG is not stable enough, this signal processing step in certainembodiments cannot be applied. Thus, in certain embodiments, this DCoffset is stabilized, for example by one or more of the following:stabilizing the temperature of the coil, ensuring that the coil is woundsymmetrically, minimizing the length of the pigtails coming out of thecoil, and/or placing the pigtails in close proximity to minimizetemperature gradients between them. Another possible method for reducingthe DC offset due to the backscattered signals is to use the methoddescribed in I. P. Giles, J. Mackintosh, J. McMillan, and B. Culshaw,“Coherent backscatter-induced drift in phase-modulated optical fibergyroscopes,” Electron. Lett., Vol. 22, No. 9, 494-496 (1986), whichinvolves properly selecting the modulation amplitude of the phasemodulation such that the main components of this spurious signal arecancelled.

In certain embodiments, a fiber-optic sensor is operated by providing afiber-optic sensor comprising an optical fiber coil having an opticallength and a laser source optically coupled to the coil. The lasersource has a coherence length such that the sensor has a phase noisebelow a predetermined value, and the coherence length is less than theoptical length. Operating the fiber-optic sensor further comprisesstabilizing a DC offset of the sensor, for example, using one or more ofthe techniques mentioned above. Light from the source to the coil istransmitted as a first signal and a second signal. The first signalpropagates along the coil in a first direction and the second signalpropagates along the coil in a second direction opposite to the firstdirection, with the optical paths of the first signal and the secondsignal are substantially reciprocal with one another. The first signaland the second signal are then combined together to generate a thirdsignal, which is detected to measure the perturbation of the sensor(e.g., the rotation of the FOG).

It is clear from FIGS. 4 and 5 that another way to strike a satisfactorycompromise between backscattering noise and DC offset in certainembodiments is to reduce the coherence length of the source below theloop length. By doing so, FIG. 4 indicates that the noise decreasesrapidly (roughly proportionally to the coherence length), while FIG. 5shows that the DC offset also decreases rapidly (also roughlyproportionally to the coherence length). A preferred mode of operationin certain embodiments described herein therefore includes operating onthe left side of the curves in FIG. 4, without reducing the coherencelength so much that it becomes so short the mean wavelength stabilitystarts to suffer again, as it does in an SFS for example. FIG. 4 showsthat in the SMF-28 FOG used here as an example, a coherence length inair of ˜9.5 m or less (linewidth of ˜10 MHz or more) will produce abackscattering phase noise of ˜0.85 μrad/√Hz or less. It can beextrapolated from FIG. 5 that it will produce a phase offset (or error)of ˜4 wad or less. Similarly, a coherence length in air of ˜0.95 m(linewidth of ˜100 MHz) will produce a backscattering phase noise of˜0.2 μrad/√Hz or less and a DC offset below 1 wad.

Some of these results were confirmed experimentally by testing thesensor 200 of FIG. 2 in accordance with certain embodiments describedherein with three lasers of increasingly long coherence lengths, namelya DFB laser (˜10-MHz linewidth, or a coherence length in silica of 6.5meters), a tunable laser (200 kHz, or a coherence length in silica of325 meters), and DFB laser (15 kHz, or a coherence length in silica of4.3 kilometers). The dependence of the backscattering noise on thesource coherence length was verified experimentally in two fiber opticgyroscopes, one made with 235 m of Corning's SMF-28 fiber, the other onewith 235 m of HC-1550-02 air-core fiber from NKT Photonics in Denmark.Both fibers were quadrupolar wound on a mandrel of 8.5-cm diameter. ThePBF gyroscope had the configuration shown in FIG. 2. The SMF-28gyroscope used the same configuration, except that the main components(the polarizer, the 3-dB coupler, and the phase modulator) werefabricated on a conventional monolithic LiNbO₃ planar structureutilizing optical waveguides. For each gyroscope, the dependence of thephase noise at the output of the gyroscope was measured as a function ofintegration time in the lock-in amplifier (see FIG. 2) for lasers withdifferent coherent lengths (or linewidth). The slope of the noisedependence on square root of integration time gave the random walk ofthe gyroscope. For comparison, the noise was also measured in these twogyroscopes with an Er-doped superfluorescent fiber source (SFS). In allfour measurements the power of the sources were adjusted such that theoutput power of the FOG was the same.

Experimental results are plotted in FIG. 4 as individual dots for eachof these lasers. FIG. 4 shows the three data points measured for theSMF-28 gyroscope with different lasers by asterisk symbols (*). For theSMF-28 FOG, the noise is only slightly higher than predicted by themodel for the two shortest coherence lengths. These two data pointsconfirm the quantitative prediction of how much the noise increases withincreasing coherence length. For comparison, the random walk (RW)measured with the SFS was 1.2 μrad/√Hz. This confirms experimentally thetheoretical prediction that for a gyroscope operated with a laser ofmoderate coherence, namely a coherence length of 6.5 meters for a looplength of 235 meters in this particular embodiment, the gyroscope's RWcan be very close to that of the same gyroscope operated conventionallywith a broadband source. By using a laser with a slightly shortercoherence length (a meter for example, a few tens of centimeters, or afew centimeters), the RW would be even smaller, and the sensitivity ofthe FOG would be improved over the sensitivity of the same gyroscopeinterrogated with a broadband source. FIG. 4 shows that when thecoherence length was increased to 4.3 kilometers, the noise did notdecrease below the value measured with the 325-meter coherence length,unlike expected from the model. When performing these measurements, wealso observed increased long-term variations in the output of thegyroscope, which had to be filtered out mathematically to infer thenoise. These results suggest that the particular laser that was usedmight have had a higher phase and intensity noise than expected.

The long-term drift of the gyroscope output was also observed toincrease significantly as the coherence length of the source isincreased. For example, with the 10-MHz laser, the long-term driftmeasured over a period of a few hours was about ±7.5 wad for a 1-sintegration time, whereas it was only ±0.8 wad when this gyroscope wasoperated with the SFS and the same integration time. This long-termdrift was observed to increase significantly with both the 200-kHzlinewidth laser and the 15-kHz linewidth source. These measurements areconsistent with the general trend predicted by FIG. 5: as the linewidthdecreases, the DC offset due to the mean value of the backscatteredsignal increases, and it eventually reaches a maximum when the coherencelength greatly exceeds the loop length.

FIG. 4 shows similar results for the air-core FOG tested with two of thelasers, with the data point shown as a plus symbol (+). The maindifference is that a reduction in the RW noise at longer coherencelengths was actually observed, presumably because the backscatteringnoise of the gyroscope dominated over other additional internal noise inthe source itself. The experimental points are higher than the predictedbest-case scenario, but show the predicted trend.

As described herein, simulations have shown that the coherentbackscattering noise of a fiber optic gyroscope interrogated with alaser in accordance with certain embodiments described herein decreaseswhen the coherence length of the laser is increased above the length ofthe sensing loop. This decrease is relatively slow, roughly as thesquare root of the laser linewidth, but sufficient to reduce thebackscattering noise below the level of the excess noise in the samegyroscope interrogated conventionally with a broadband source. The firstexperimental evidence of this reduction in a conventional gyroscope hasbeen obtained using a 235-m length of air-core fiber in accordance withcertain embodiments described herein. This new principle can beadvantageously used for a fiber optic gyroscope made with an air-corefiber in accordance with certain embodiments described herein, which isessentially immune to the Kerr effect but exhibits strongerbackscattering than a conventional fiber. It is also applicable to aconventional gyroscope, in which the Kerr-induced drift can beeliminated by other means, for example with a square-wave modulation ofthe laser amplitude in accordance with certain embodiments describedherein. This approach offers for the first time the potential of a fiberoptic gyroscope with a shot-noise-limited detection, which willtranslate into a greater sensitivity.

In certain embodiments, in which the sensor has a phase noise whichvaries as a function of coherence length, the phase noise has a peakvalue at a predetermined value of coherence length (e.g., at or near thelength of the coil), as shown in FIG. 4. In certain such embodiments,the coherence length is selected such that the phase noise is at least afactor of two less than the peak value of the phase noise. In certainembodiments, the coherence length of the laser source is less than thepredetermined value. In certain embodiments, the coherence length isselected such that the sensor has a phase noise less than or equal to apredetermined value. For example, in certain embodiments, the coherencelength of the narrowband source is selected such that the sensor has aphase noise less than it would have if driven by a broadband source.Typically, the phase noise resulting when the sensor is driven by abroadband source (e.g., an Er-doped SFS) is about 1 μrad/√Hz, or is in arange between about 0.5 μrad/√Hz and about 2 μrad/√Hz. Thus, in certainembodiments, the coherence length of the narrowband source is selectedto result in a phase noise less than about 1 μrad/√Hz, less than about0.5 μrad/μHz, or less than about 2 μrad/√Hz.

Various embodiments have been described above. Although this inventionhas been described with reference to these specific embodiments, thedescriptions are intended to be illustrative of the invention and arenot intended to be limiting. Various modifications and applications mayoccur to those skilled in the art without departing from the true spiritand scope of the invention as defined in the appended claims.

What is claimed is:
 1. An optical sensor comprising: an optical loop;and a laser source optically coupled to the loop, the laser sourcehaving a coherence length, wherein light from the laser source istransmitted to the loop as a first signal and a second signal, whereinthe first signal and the second signal counterpropagate along opticalpaths that are substantially reciprocal with one another and the firstsignal and the second signal are combined together aftercounterpropagating through the loop to generate a third signal, whereinthe coherence length is greater than 1 meter or is in a range between200 microns and 10 centimeters.
 2. The sensor of claim 1, wherein thecoherence length is in a range between 500 microns and 10 centimeters.3. The sensor of claim 1, wherein the coherence length is in a rangebetween 1 millimeter and 10 centimeters.
 4. The sensor of claim 1,wherein the coherence length is in a range between 1 millimeter and 1centimeter.
 5. The sensor of claim 1, wherein the coherence length is ina range between 1 centimeter and 10 centimeters.
 6. The sensor of claim1, wherein the coherence length is in a range between 1 meter and 1kilometer.
 7. The sensor of claim 1, wherein the coherence length is ina range between 1 meter and 100 meters.
 8. The sensor of claim 1,wherein the coherence length is in a range between 10 meters and 100meters.
 9. The sensor of claim 1, wherein the coherence length is in arange between 10 meters and 1 kilometer.
 10. The sensor of claim 1,wherein the coherence length is in a range between 100 meters and 10kilometers.
 11. The sensor of claim 1, wherein the coherence length isless than a length of the loop.
 12. The sensor of claim 1, wherein thesensor has a phase noise which varies as a function of coherence length,the phase noise having a peak value at a predetermined value ofcoherence length, wherein the phase noise for the coherence length is atleast a factor of two less than the peak value of the phase noise. 13.The sensor of claim 1, wherein the sensor has a phase noise which variesas a function of coherence length, the phase noise having a peak valueat a predetermined value of coherence length, wherein the coherencelength of the laser source is less than the predetermined value.
 14. Thesensor of claim 1, wherein the sensor has a phase noise which varies asa function of coherence length, wherein the coherence length results ina phase noise less than about 2 μrad/√Hz.
 15. The sensor of claim 1,wherein the sensor has a phase noise which varies as a function ofcoherence length, wherein the coherence length results in a phase noiseless than about 1 μrad/√Hz.
 16. The sensor of claim 1, wherein thesensor has a phase noise which varies as a function of coherence length,wherein the coherence length results in a phase noise less than about0.5 μrad/√Hz.
 17. The sensor of claim 1, wherein the first signal andthe second signal have the same frequency as the light from the lasersource.
 18. The sensor of claim 1, wherein the laser source has a meanwavelength stability greater than 1 part per million.
 19. The sensor ofclaim 1, wherein the sensor is a fiber-optic gyroscope and wherein theloop comprises a standard Sagnac loop which comprises an optical fibercoil.
 20. A method of operating an optical sensor, the methodcomprising: providing an optical sensor comprising an optical loophaving a length; stabilizing a DC offset of the sensor; receiving lightby the loop from a laser source optically coupled to the loop as a firstsignal and a second signal, the first signal and the second signalcounterpropagating along optical paths that are substantially reciprocalwith one another, the laser source having a coherence length such thatthe sensor has a phase noise below a predetermined value, wherein thecoherence length is less than the length; and combining the first signaland the second signal together to generate a third signal.
 21. Themethod of claim 20, wherein a ratio of the coherence length to thelength of the loop is greater than 0.1.
 22. The method of claim 20,wherein the coherence length is greater than 1 meter or is in a rangebetween 200 microns and 10 centimeters.
 23. A method of configuring anoptical sensor, the method comprising: providing an optical sensorcomprising an optical loop, the sensor having a phase noise which variesas a function of coherence length of a laser source; receiving light bythe loop from the laser source as a first signal and a second signal,the first signal and the second signal counterpropagating along opticalpaths that are substantially reciprocal with one another, the lasersource having a coherence length such that the sensor has a phase noiseless than a phase noise resulting when the sensor is driven by abroadband source; and combining the first signal and the second signaltogether after counterpropagating through the loop to generate a thirdsignal.
 24. The method of claim 23, wherein the coherence length is lessthan a length of the loop.
 25. The method of claim 24, wherein a ratioof the coherence length to the length of the loop is greater than 0.1.26. The method of claim 23, wherein the coherence length is greater than1 meter or is in a range between 200 microns and 10 centimeters.
 27. Themethod of claim 23, wherein the coherence length results in a phasenoise less than about 2 μrad/√Hz.
 28. The method of claim 23, whereinthe coherence length results in a phase noise less than about 1μrad/√Hz.
 29. The method of claim 23, wherein the coherence lengthresults in a phase noise less than about 0.5 μrad/√Hz.
 30. The method ofclaim 23, wherein the first signal and the second signal have the samefrequency as the light from the laser source.